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Full text of "Treatise On Analysis Vol-Ii"

176       XIII    INTEGRATION

(f) Suppose that X is compact and that the measure ft is diffuse (13.18). Show that,
for every neighborhood V of 0 in <?"(X, ft) and every function / e ^(X, ft), there
exists an integer n such that all the functions af (where a is any real number) belong to
y _j_ . . . _|_ v (n summands). Deduce that every continuous linear form on ^(X, ft)
is identically zero, and hence that every vector subspace of finite codimension in
, ft) is dense in <^(X, ft).

3.    Suppose that X is compact and the measure ft diffuse (1 3.18).

(a)    Let (fynzi be a Hilbert basis of L^(X, ft). Show that, for each real number
8 > 0, there exists a compact subset Y of X such that ft(X  Y) ^ 8 and the sequence
(/n)n2 is total in LR(, ftY). (By using Problems 2(e) and 2(f), show that there exists
a sequence of linear combinations of the / (n ^> 2) which converges in measure to /i ;
then use Problem 2(c) and EgorofF s theorem.)

(b)    Show that there exists a bounded measurable function h such that the sequence
(%fn)nZ2 is total in Lj(X, ft). (Choose h > 0 such that/i/A $ JS?J(X, ft), and then show
that no nonnegligible function can be orthogonal to hfn for all n ^ 2.)

4.    Let;? be a finite real number *>l. A subset H of ^?R(X, ft) is said to be equi-integrable
if, for each e> 0, there exists a compact subset K of X such that         \f\p dp, rge

for all /e H, and a real number 8 > 0 such that f \f\p dp-^s for all /e H and all
integrable sets A of measure ft(A) < S.

(a)    On an equi-integrable set H, show that the topology of convergence in measure
is the same as that defined by the seminorm Np . Is the conclusion true if H is merely
bounded in -S?p?

(b)   A sequence (/) in -S?fc(X, ft) is convergent if and only if it is equi-integrable and
convergent in measure.

(c)    Suppose that the measure ft is bounded and that ft(X) = 1 . Let (/) be a sequence

of functions belonging to J2P, and suppose that lim \fndfji= lim \\&fn\ dft = l,

n-*ooj                n-ooj

lim Nid - i/n|) = 0. Show that lim f | Sfn\ dp. = 0. (Use (b).)

n-+oo                                                           n-fooj

(d)   With ft as in (c), let (/) be a sequence of functions belonging to & c suc^ tnat

lim f/n</ft=lim f|/n|^ lim f|/;|1/2 dp,= 1.

n-*oo J                  n-+oo J                      n-*ooj

Show that HmNi(l /n) = 0. (Reduce to the situation of (c) above by using the

n-+oo

Cauchy-Schwarz inequality; write

H - fn\^\l- \fn\\ + \\fn\-fn\

and

5.   Let F be a real-valued function with period 1 on R which is integrable on the unit
interval I = [0, 1 ] (with respect to Lebesgue measure).
(a)   If /is any bounded measurable function on [0, 1], then

lim f /(/)FOi/) dt = ( f F (0 dt]( f'

-Jo                          \Jo             / woection 13.8,